HP 15c - Manual (Page 18)

18 Section 1: Using _ Effectively
18
A final consideration for this example is to choose the initial estimates that would be
appropriate. Because the argument of the inverse cosine must be between −1 and 1, x must be
more negative than about −0.1059 (found by trial and error or by using _). The initial
guesses might be near but more negative than this value, −0.11 and −0.2 for example.
(The complex equation used in this example is solved using an iterative procedure in the
example on page 69. Another method for solving a system of nonlinear equations is
described on page 102.)
Finding Local Extremes of a Function
Using the Derivative
The traditional way to find local maximums and minimums of a function's graph uses the
derivative of the function. The derivative is a function that describes the slope of the graph.
Values of x at which the derivative is zero represent potential local extremes of the function.
(Although less common for well-behaved functions, values of x where the derivative is
infinite or undefined are also possible extremes.) If you can express the derivative of a
function in closed form, you can use _ to find where the derivative is zero—showing
where the function may be maximum or minimum.
Example: For the design of a vertical broadcasting tower, radio engineer Ann Tenor wants to
find the angle from the tower at which the relative field intensity is most negative. The
relative intensity created by the tower is given by
 
  
sin )] 2 cos( 1 [
) 2 cos( ) cos 2 cos(
h
h h
E



where E is the relative field intensity, h is the antenna height in wavelengths, and θ is the
angle from vertical in radians. The height is 0.6 wavelengths for her design.
The desired angle is one at which the derivative of the intensity with respect to θ is zero.
To save program memory space and execution time, store the following constants in registers
and recall them as needed:
r
0
= 2πh and is stored in register R
0
,
r
1
= cos(2πh) and is stored in register R
1
,
r
2
= 1/[1 − cos(2πh)] and is stored in register R
2
.
The derivative of the intensity E with respect to the angle θ is given by



 
 
 



tan sin
) cos cos(
) cos sin( 1 0
0 0 2
r r
r r r
d
dE
.
Key in a subroutine to calculate the derivative.